Slope fields are used to give a general idea of the shape of a graph f(x)+c where c is any real number without actually solving the differential equation. The differential equation gives slope at any point (x,y) and we can use this to plot the slopes at any point (x,y). After plotting the various slopes, a general idea of the shape of the graph is gained.
 |
| frac {dy}{dx} = 2x + y |
| (x,y) |
y' |
(x,y) |
y' |
(x,y) |
y' |
(x,y) |
y' |
||
| (3,3) |
9 |
(-3,3) |
-3 |
(-3,-3) |
-9 |
(3,-3) |
3 |
||
| (3,2) |
8 |
(-3,2) |
4 |
(-3,-2) |
-8 |
(3,-2) |
4 |
||
| (3,1) |
7 |
(-3,1) |
-5 |
(-3,-1) |
-7 |
(3,-1) |
5 |
||
| (2,3) |
7 |
(-2,3) |
-1 |
(-2,-3) |
-7 |
(2,-3) |
1 |
||
| (2,2) |
6 |
(-2,2) |
-2 |
(-2,-2) |
-6 |
(2,-2) |
2 |
||
| (2,1) |
5 |
(-2,1) |
-3 |
(-2,-1) |
-5 |
(2,-1) |
-3 |
||
| (1,1) |
3 |
(-1,1) |
-1 |
(-1,-1) |
-3 |
(1,-1) |
1 |
||
| (1,2) |
4 |
(-1,2) |
0 |
(-1,-2) |
-4 |
(1,-2) |
0 |
||
| (1,3) |
5 |
(-1,3) |
1 |
(-1,-3) |
-5 |
(1,-3) |
-1 |